The invariants of the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$

TL;DR

This paper studies the invariants of the Weil representation associated with positive-definite even lattices, revealing they are generated from five fundamental invariants and applying this to simplify the description of certain Jacobi forms.

Contribution

It identifies the invariants of the Weil representation as being induced from five fundamental invariants, providing new insights into the structure of vector valued theta functions.

Findings

Invariants are induced from five fundamental invariants.

Simplifies generating sets for Jacobi forms of singular weight.

Provides a structural understanding of the Weil representation's invariants.

Abstract

The transformation behaviour of the vector valued theta function of a positive-definite even lattice under the metaplectic group $\mathrm{Mp}_2(\mathbb{Z})$ is described by the Weil representation. We show that the invariants of this representation are induced from $5$ fundamental invariants. As an application we give simple generating sets for Jacobi forms of singular weight.